Abstract Error Groups Via Jones Unitary Braid Group Representations at q=i

TL;DR

This paper classifies certain abstract groups related to quantum error correction, showing their connection to Jones braid group representations at q=i, and explores their potential for new quantum error correction codes.

Contribution

It introduces a new classification of error groups via Jones braid group representations at q=i, expanding the understanding of quantum error correction structures.

Findings

Abstract error groups are not isomorphic to Pauli groups.

Corresponding error bases are equivalent to Pauli bases modulo phase.

Extension groups are finite images of Jones braid group representations at q=i.

Abstract

In this paper, we classify a type of abstract groups by the central products of dihedral groups and quaternion groups. We recognize them as abstract error groups which are often not isomorphic to the Pauli groups in the literature. We show the corresponding nice error bases equivalent to the Pauli error bases modulo phase factors. The extension of these abstract groups by the symmetric group are finite images of the Jones unitary representations (or modulo a phase factor) of the braid group at q=i or r=4. We hope this work can finally lead to new families of quantum error correction codes via the representation theory of the braid group.